If you go with option A you'll probably be fine. It seems to be very common for starters to be made with only DME and water, and the beers seem to come out just fine.
But, Although there is some Free Amino Nitrogen (FAN) in malt extract, there is not enough for growth of the entire biomass, making a FAN limited batch culture. The result is cells that have less amino acids which means less enzymes and proteins that it will use to metabolize the wort into beer. One the yeast is in the new wort it will assimilate the new FAN source and make up for the difference, but it will be a slower start, and the yeast will lag a little at the end of fermentation. FAN limitation will also stop new cell division from starting because there will be no nitrogen to synthesize new DNA.[1]
Honestly, even though I know better, I've done most of my beers without adding yeast nutrient to the starter and they came out fine. However, your safest option would be to boil and cool the yeast nutrient and add it to the starter. It's probably just getting to the point where it could use the extra FAN.
[1]Schulze, Ulrik, et al. "Physiological effects of nitrogen starvation in an anaerobic batch culture of Saccharomyces cerevisiae." Microbiology 142.8 (1996): 2299-2310.